If I said "ポーランドに住んでいて、学生です。" would that be correct? I'm a beginner and I'm not sure if "sunde ite" sounds right to me
I don’t think you should see it as a "double て-form."
A verb in its て-form followed by certain other (helper) verbs such as います, あります, 行きます, 来ます, etc. is a common structure in Japanese. When you use this in various grammatical functions, you change the form of the second verb. The second て-form here is just one such inflection.
It’s probably more natural to say 学生で、ポーランドに住んでいます to put your status first, though.